Differentiable Definition and 290 Threads

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp.
More generally, for x0 as an interior point in the domain of a function f, then f is said to be differentiable at x0 if and only if the derivative f ′(x0) exists. In other words, the graph of f has a non-vertical tangent line at the point (x0, f(x0)). The function f is also called locally linear at x0 as it is well approximated by a linear function near this point.

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  1. L

    Discontinuous partial derivatives example

    $$f(x,y)=\left\{\begin{array}{ccc} (x^2+y^2)\sin\left(\frac{1}{\sqrt{x^2+y^2}}\right) & , & (x,y)\neq (0,0) \\ 0 & , & (x,y)=(0,0) \end{array}\right.$$ This function is differentiable at (0,0) point but ##f_x## and ##f_y## partial derivatives not continuous at (0,0) point. I need another...
  2. M

    Limiting formula for differentiable function

    For this problem and solution, I'm confused how ##x \in (c - \delta, c + \delta)## is the same as ##0 <| x - c| <\delta##. I think it is the same as ##c - \delta < x < c + \delta## which we break into parts ##c - \delta < x \implies \delta > -(x - c)## and ##x < c + \delta \implies x - c <...
  3. M

    Proof given ##x < y < z## and a twice differentiable function

    For this problem, My proof is Since ##f'## is increasing then ##x < y <z## which then ##f(x) < f(y) < f(z)## This is because, ##f''(t) \ge 0## for all t ## \rightarrow \int \frac{df'}{dt} dt \ge \int 0~dt = 0## for all t ##\rightarrow \int df' \geq 0## for all t ##f ' \geq 0## for all t...
  4. M

    Differentiable function proof given ##f''(c) = 1##

    For this problem, I'm confused by the implication from the antecedent ##0 < |x - c| < \delta## to the consequent. Should the consequent not be ##|f''(x) - f''(c)| < \frac{1}{2}## where ##\epsilon = \frac{1}{2}## (Since we are applying the definition of a limit for the first derivative curve)...
  5. M

    Proving piecewise function is k-differentiable

    For this problem, My solution is, ##F(x)=\left\{\begin{array}{ll} e^{-\frac{1}{x}} & \text { if } x>0 \\ 0 & \text { if } x \leq 0\end{array}\right.## The we differentiate both sub-function of the piecewise function. Note I assume differentiable since we are proving a result that the function...
  6. M

    True or false problem for double differentiable function

    For this true or false problem, My solution is, With rearrangement ##\frac{f(x) - f(a)}{x - a} > f'(a)## for ##x < a## since ##f''(x) > 0## implies ##f'(x)) > 0## from integration. ##f'(x) > 0## is equivalent to ##f(x)## is strictly increase which means that ##\frac{f(x) - f(a)}{x - a} > f'(a)...
  7. P

    A Vector calculus - Prove a function is not differentiable at (0,0)

    ##f\left(x\right)=\begin{cases}\sqrt{\left|xy\right|}sin\left(\frac{1}{xy}\right)&xy\ne 0\\ 0&xy=0\end{cases}## I showed it partial derivatives exist at ##(0,0)##, also it is continuous as ##(0,0)## but now I have to show if it differentiable or not at ##(0,0)##. According to answers it is not...
  8. redtree

    B Difference between a continuously differentiable function and a wave

    What is the difference between an absolutely continuously differentiable function and a wave? Are all absolutely continuously differentiable equations waves?
  9. M

    MHB Is f Differentiable at (0,0)?

    Hey! :giggle: We consider the function\begin{align*}f:\mathbb{R}^2 &\rightarrow\mathbb{R} \\ (x,y)&\mapsto \begin{cases}\frac{x^3}{x^2+y^2} & \text{ if } (x,y)\neq(0,0) \\ 0 & \text{ if } (x,y)=(0,0) \end{cases}\end{align*} (a) Show that all directional derivatives of $f$ in $(0,0)$ exist...
  10. M

    MHB Piecewise function: differentiable but not continuously differentiable

    Hey! :giggle: We have the function $$f(x,y)=\begin{cases}x^2\sin\left (\frac{1}{x}\right )+y^2\sin\left (\frac{1}{y}\right ) & \text{ if } xy\neq 0 \\ x^2\sin\left (\frac{1}{x}\right ) & \text{ if } x\neq 0, y=0 \\ y^2\sin\left (\frac{1}{y}\right ) & \text{ if } x= 0, y\neq 0 \\ 0 & \text{ if...
  11. M

    MHB F is partially differentiable in (0,0) but not total differentiable

    Hey! :giggle: Let's consider the function \begin{align*}f:\mathbb{R}^2&\rightarrow \mathbb{R} \\ (x,y)&\mapsto \sqrt{|x|\cdot |y|}\end{align*} Show that $f$ is partially differentiable in $(0,0)$ but not total differentiable.I have done the following: We prove that $f$ is partially...
  12. D

    How to show a function is twice continuously differentiable?

    this seems to come up frequently in undergrad math classes so it is worth asking, what is the simplest and most efficient way to show ##f(x)\in C^2(\mathbb{R})## given $$f(x)=\begin{cases} (x+1)^4 & x<-\frac{1}{2} \\ 2x^4-\frac{3}{2}x^2+\frac{5}{16} & -\frac{1}{2}\leq x \end{cases}$$ And what...
  13. T

    Determine whether a function is continuous or differentiable

    Perhaps use the definition of continuity, partial differentiability?
  14. Mayhem

    I Showing that a set of differentiable functions is a subspace of R

    Problem: Show that the set of differentiable real-valued functions ##f## on the interval ##(-4,4)## such that ##f'(-1) = 3f(2)## is a subspace of ##\mathbb{R}^{(-4,4)}## This is my first bouts with rigorous mathematics and my brain is not at all wired for attacking problems like this (yet). I...
  15. LCSphysicist

    I Square of a differentiable functional

    I will consider first the case of ## \left [ J \right ] = \int f(x,y,y') ##, if it is right believe is easy to generalize... $$ \Delta J $$ $$\int (f(x,y+h,y'+h'))^2 - (f(x,y,y'))^2 $$ $$\int \sim [f(x,y,y') + f_{y}(x,y,y')h + f_{y'}(x,y,y')h']^2 - [f(x,y,y')]^2$$ to first order: $$\int \sim...
  16. hilbert2

    A Softened potential well / potential step

    Do any of you know of an article or book chapter that discusses the difference between a discontinuous potential well of length ##2L## ##V(x)=\left\{\begin{array}{cc}0, & |x-x_0 |<L\\V_0 & |x-x_0 |\geq L\end{array}\right.## and a differentiable one ##\displaystyle V(x) = V_0...
  17. C

    I An interesting point regarding critical points and extrema

    Hi all, I have recently faced some problem about distances between two curves, and (re?)"discovered" an interesting point that I would like to share with you. In the following, we consider a function of two variables ##f(x,y)##, but it should be clear that the definitions and the result is...
  18. sergey_le

    Let function ƒ be Differentiable

    What I've tried is: I have defined a function g(x)=f(x)-x^2/2. g Differentiable in the interval [0,1] As a difference of function in the interval. so -x≤g'(x)≤1-x for all x∈[0,1] than -1≤g'(x)≤0 or 0≤g'(x)≤1 . Then use the Intermediate value theorem . The problem is I am not given that f' is...
  19. sergey_le

    Multi-Choice Question: Differentiable function

    be f Differentiable function In section [0,1] and f(0)=0, f(1)=1. so: a. f A monotonous function arises in section [0,1]. b. There is a point c∈[0,1] so that f'(c)=1. c. There is a point c∈(0,1) where f has Local max. I have to choose one correct answer.
  20. W

    I Differentiable manifolds over fields other than R, C

    [Moderator's note: Spin-off from another thread.] You need the structure of a topological vector field K with 0 as a limit point of K-{0}. The TVF structure allows the addition and quotient expression to make sense; you need 0 as a limit point to define the limit as h-->0 and the topology to...
  21. J

    I How do charts on differentiable manifolds have derivatives without a metric?

    I was reading about differentiable manifolds on wikipedia, and in the definition it never specifies that the differentiable manifold has a metric on it. I understand that you can set up limits of functions in topological spaces without a metric being defined, but my understanding of derivatives...
  22. M

    Prove the dipole potential is differentiable everywhere except at the surface

    The dipole potential is given by: ##\displaystyle\psi=\int_{V'} \dfrac{\rho}{|\mathbf{r}-\mathbf{r'}|} dV' +\oint_{S'} \dfrac{\sigma}{|\mathbf{r}-\mathbf{r'}|} dS'## I need to prove that ##\psi## is differentiable at points except at boundary ##S'## (where it is discontinuous) I know if...
  23. Opressor

    A What is a differentiable variety?

    In mathematics, variety is a generalization of the surface idea. There are several types of varieties, according to the properties they possess. The most usual are the topological varieties and the differentiable varieties. but I still do not know what it is!
  24. cianfa72

    B Differentiable function - definition on a manifold

    Hi, a basic question related to differential manifold definition. Leveraging on the atlas's charts ##\left\{(U_i,\varphi_i)\right\} ## we actually define on ##M## the notion of differentiable function. Now take a specific chart ##\left(U,\varphi \right)## and consider a function ##f## defined...
  25. Eclair_de_XII

    Show that a continuously differentiable function is not 1-1

    Homework Statement "Let ##f:ℝ^2\rightarrow ℝ## be a continuously differentiable function. Show that ##f## is not one-to-one." Homework Equations A function ##f:ℝ^n\rightarrow ℝ^m## is continuously differentiable if all the partial derivatives of all the components of ##f## exist and are...
  26. opus

    Show that this graph isn't differentiable at x=1

    Homework Statement Sketch the graph of the following function and use the definition of the derivative to show that the function is not differentiable at x=1. $$f(x) = \begin{cases} {-x^2+2} & \text{if } x \leq 1 \\ x & \text{if } x > 1 \end{cases}$$ Homework Equations Derivative: $$f'(x) =...
  27. Eclair_de_XII

    Showing that f(x,y) = √|xy| is not differentiable at (0,0)

    Homework Statement "Let ##f:ℝ^2\rightarrow ℝ## be defined by ##f(x,y)=\sqrt{|xy|}##. Show that ##f## is not differentiable at ##(0,0)##." Homework Equations Differentiability: If ##f:ℝ^n\rightarrow ℝ^m## is differentiable at ##a\in ℝ^n##, then there exists a unique linear transformation such...
  28. T

    Where is ##(z+1)Ln(z)## differentiable?

    Homework Statement Find the domain in which the complex-variable function ##f(z)=(z+1)Ln(z)## is differentiable. Note: ##Ln(z)## is the principal complex logarithmic function. Homework Equations Cuachy-Riemann Equations? The Attempt at a Solution The solution I have in mind would be to let...
  29. F

    What it means to be infinitely differentiable

    <Moderator's note: Moved from a homework forum.> 1. Homework Statement I am wondering if it is the same to say : Function f is infinitely many times continuously differentiable AND Function f is infinitely many times differentiable And if it is not the same, which one defines: C^infinity...
  30. evinda

    MHB Show that solution is infinitely many times differentiable

    Hello! (Wave) I want to prove that the solution $u$, $u(x,t)=\frac{1}{2 \sqrt{k \pi t}} \int_{-\infty}^{+\infty} e^{-\frac{(x-s)^2}{4kt}} \phi(s) ds, x \in \mathbb{R}, t>0 (\star)$, of the initial value problem for the heat equation, with continuous and bounded initial value $\phi$, is...
  31. J

    Is this function continuous and differentiable?

    Homework Statement Homework Equations Solve using limits. Function is continuous if it's graph is continuous throughout. here the (x-1) term gets canceled in numerator and denominator. So we have a continuous graph of (x+1). The Attempt at a Solution The (x-1) term gets canceled from...
  32. M

    Rule de l'Hôpital - Continuous and Differentiable

    Homework Statement Prove that ##h: [0,\infty) \rightarrow \mathbb R, x \mapsto \begin{cases} x^x, \ \ x>0\\ 1, \ \ x = 0\\ \end{cases} ## is continuous but not differentiable at x = 0. The Attempt at a Solution To show continuity, the limit as x approaches 0 from the right must equal to 1...
  33. M

    Which points in the domain of f are differentiable and what is their derivative?

    Homework Statement Let f be ##f:[0,\infty]\rightarrow \mathbb R \\ f(x):= \begin{cases} e^{-x}sin(x), \ if \ \ x\in[2k\pi,(2k+1)\pi] for \ a \ k \in \mathbb N_0 \\ 0 \ \ otherwise\\ \end{cases}## Exercise: Determine all inner points of the domain where f is also differentiable and determine...
  34. M

    MHB Continuously differentiable series

    Hey! :o I want to show that series $$f(x)=\sum_{k=1}^{\infty}2^k\sin (3^{-k}x)$$ is continuously differentiable. We have that $|2^k\sin (3^{-k}x)|\leq 2^k\cdot 3^{-k}=\left (\frac{2}{3}\right )^k$, or not? The sum $\sum_{k=1}^{\infty}\left (\frac{2}{3}\right )^k$ converges as a geometric...
  35. G

    Prove that this function is holomorphic

    Homework Statement Prove that the function ## f(z)= 1/\sqrt{2}(\sqrt{\sqrt{x^{2}+y^{2}}+x}+i*sgn(y)\sqrt{\sqrt{x^{2}+y^{2}}-x})## is holomorphic on the domain ## \Omega = \left \{ z: z \neq 0, \left | \arg{z} \right | <\pi\right \} ## and further that in this domain ##f(z)^{2} = z. ##...
  36. SemM

    A Can Quantum Mechanics be Studied in Banach Spaces or Other Non-Hilbert Spaces?

    Hi, are there any models known in QM where the wavefunctions do not have to be infinitely differentiable, and thus can exist in other spaces than the Hilbert space? I assume Banach spaces allow elements that are not infinitely differentiable as subsets. Can therefore certain phenomena in QM be...
  37. J

    Continous and differentiable function

    Homework Statement Homework Equations All polynomial functions are continuous so the function is continuous everywhere. For differnetiable we differentiate the polynomial. But how to do this? The Attempt at a Solution I only removed option A as polynomial functions are always continuous. How...
  38. lfdahl

    MHB Determine all real valued differentiable functions f(x)+f(y)=f(xy)

    Determine, with proof, all the real-valued differentiable functions $f$, defined for real $x > 0$, which satisfy $f(x) + f(y) = f(xy)$ for all $x, y > 0$.
  39. Oats

    Show that this function is differentiable

    Homework Statement [/B] 2. The attempt at a solution I'm not really sure where to start. We just want to show that ##\lim_{x \to c} \frac{f(x) - f(c)}{x - c} = 0##. I see that ##\lim_{x \to c} (x - c)^2 = 0##. I feel that this may be a simple trick of inequalities, but I am having a complete...
  40. FallArk

    MHB Need help, are these functions differentiable?

    I want to figure out whether the functions are differentiable at c. I think I should use some of the trig identities, but I'm not sure which ones. Any tips?
  41. M

    MHB Exploring the Global Maximum of a Differentiable Function

    Hey! :o I am looking at the following: The differentiable and so also continuous function $f:[a,b]\rightarrow \mathbb{R}$ gets its global maximum at a point $x^{\star}$. Show that the following holds $$f'(x^{\star})=\left\{\begin{matrix} =0\\ \leq 0\\ \geq 0 \end{matrix}\right. , \ \text{...
  42. J

    Find A and B so that F(x) is a Differentiable Function

    Homework Statement Find the values of a and b that make f a differentiable function. Note: F(x) is a piecewise function f(x): Ax^2 - Bx, X ≤ 1 Alnx + B, X > 1 Homework EquationsThe Attempt at a Solution Made the two equations equal each other. Ax^2 - Bx = Alnx + B Inserting x=1 gives, A - B =...
  43. lep11

    Expressing defined integral as composition of differentiable functions

    Homework Statement Let ##f(t)=\int_{t}^{t^2} \frac{1}{s+\sin{s}}ds,t>1.##Express ##f## as a composition of two differentiable functions ##g:ℝ→ℝ^2## and ##h:ℝ^2→ℝ##. In addition, find the derivative of ##f## (using the composition). Homework EquationsThe Attempt at a Solution Honestly, I have...
  44. lep11

    Prove a statement regarding differentiable mv-function

    Homework Statement Let ##f##: ##G\subset\mathbb{R}^2\rightarrow\mathbb{R}## be differentiable at ##(x_0,y_0)\in{G}## and ## \lim_{(x, y) \to (x_0, y_0)} \frac{f(x,y) -a -b(x-x_0) -c(y-y_0)}{\sqrt{(x-x_0)^{2} + (y-y_0)^{2}}} = 0.## The task is to prove that then ##a=f(x_0,y_0),b=f_x(x_0,y_0)##...
  45. Y

    MHB What is the proof for f(1.1)>-0.1?

    Hello all, I am not sure how to approach this question: Let f(x) be a continuous and differentiable function of order 2. Let f''(x) >0 for all values of x. The tangent line to the function at x=1 is y=-x+1. Show that f(1.1)>-0.1. Thanks!
  46. M

    B Continuous and differentiable functions

    "If a function can be differentiated, it is a continuous function" By contraposition: "If a function is not continuous, it cannot be differentiated" Here comes the question: Is the following statement true? "If a function is not right(left) continuous in a certain point a, then the function...
  47. evinda

    MHB Is the Function $f(x,y)=\frac{x}{y}+\frac{y}{x}$ Differentiable and $C^1$?

    Hello! (Wave) Suppose that we want to check if $f(x,y)=\frac{x}{y}+\frac{y}{x}$ is differentiable at each point of its domain and if it is $C^1$. The domain is $D=\{ (x,y) \in \mathbb{R}^2: x \neq 0 \text{ and } y \neq 0\}$.The partial derivatives ...
  48. L

    B Continuous but Not Differentiable

    Suppose a certain function in continuous at c and (c. f(c)) exists, then which of the two could be false: \displaystyle \lim_{x \rightarrow c^-} {f(x)} = \lim_{x \rightarrow c^+} {f(x)}, and \displaystyle f'(c)? I feel like both could be false, because if the formal derivative at a point...
  49. M

    Is ln(x) differentiable at negative x-axis

    Since lnx is defined for positive x only shouldn't the derivative of lnx be 1/x, where x is positive. My books does not specify that x must be positive, so is lnx differentiable for all x?
  50. PcumP_Ravenclaw

    Is the function differentiable at x = p?

    Hello mates, is the function ## f(x) = \frac{(2^x - 1)}{x} ## differentiable at x = 0? For it to be differentiable it has to be continuous? From the graph f(0) is undefined although limit exists. I have read that at points like a corner, gap and vertical tangents it is not differentiable. So...
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